#6605772

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,33,39)(20,34,36,41)(21,26,22,27)(24,32,40,31)(25,37,42,35)(30,38,45,48)(43,46,47,44);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,32,23,33,21,31)(18,24,26,39,28,40,27,29)(20,30,35,44,36,45,37,46)(25,38,34,47,42,48,41,43);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,21,37,23,36,22,35)(18,25,27,41,28,42,26,34)(19,30,31,46,33,45,32,44)(24,38,29,43,40,48,39,47);(9,11)(10,13)(12,15)(14,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,41,42,34)(30,44,45,46)(38,43,48,47);(11,15)(13,16)(20,36)(25,42)(30,45)(34,41)(35,37)(38,48)(43,47)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6512968);